The curve shortening flow deforms a closed curve gamma in the direction of its normal vectors. Precisely, one deforms gamma by the evolution equation 6g6t=k n, where kappa is the curvature and n is the normal vector; the derivative of the tangent vector T with respect to arclength s is T'(s) = kappan.;In the first part of the thesis, we will study the evolution of convex, closed curve gamma in R2 . We will study the theorem of Tso, which says that gamma becomes analytic in short time, and converges to a point within finite time. We then explain the Gage theorem on the asymptotic behavior of gamma: if the area enclosed by gamma is rescaled to be a constant (e.g., pi) then gamma converges smoothly to a unit circle.;In local coordinates, the curve shortening flow (of a curve in an n-dimensional manifold M) can be written as dgkdt= 62gk6s 2+i,j=1n Gki,j6g i6s6g j6s, k=1,2,...,n;0.1 here s is the arclength parameter , we write gamma = (gamma1, gamma2, ..., gamma n) in local coordinate system; and Gkij are the Christoffel symbols of the connection on the manifold M.;In the second part of this thesis, we use the same equation (0.1), but now take s to be a fixed parameter on gamma; so s is not an arclength parameter of gamma when t > 0. This modified equation is called the one-dimensional harmonic heat flow. On general manifolds of nonpositive curvature, under the harmonic heat flow, any loop converges to a minimal geodesic loop as t → infinity. As a special case, in Euclidean spaces, all loops, convex or not, will shrink to a point. We will present a proof of this fact.
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